JEE Main Mathematics — Calculus previous year questions with solutions.
If the area of the region ${(x,y):|{x}^{2}-2|\leq y\leq x}$ is $A$, then $6A+16\sqrt{2}$ is equal to ______________
If the area of the region $S={(x,y):2y-{y}^{2}\leq {x}^{2}\leq 2y,x\geq y}$ is equal to $\frac{n+2}{n+1}-\frac{\pi }{n-1},$ then the natural number $n$ is equal to $_______$
If the area of the region bounded by the curves ${y}^{2}-2y=-x$ and $x+y=0$ is $A$, then $8A=$
If the area enclosed by the parabolas ${P}_{1}:2y=5{x}^{2}$ and ${P}_{2}:{x}^{2}-y+6=0$ is equal to the area enclosed by ${P}_{1}$ and $y=\alpha x,\alpha >0$, then ${\alpha }^{3}$ is equal to _____ .
If the area bounded by the curve $2{y}^{2}=3x$, lines $x+y=3,y=0$ and outside the circle ${(x-3)}^{2}+{y}^{2}=2$ is A, then $4(\pi +4A)$ is equal to __________.
If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{4x}{({x}^{2}-1)}y=\frac{x+2}{{({x}^{2}-1)}^{\frac{5}{2}}},x>1$ such that $y(2)=\frac{2}{9}{\mathrm{log}}_{e}(2+\sqrt{3})$ and $y(\sqrt{2})=\alpha {\mathrm{log}}_{e}(\sqrt{\alpha }+\beta )+\beta -\sqrt{\gamma },\alpha ,\beta ,\gamma \in \mathbb{N}$, then $\alpha \beta \gamma$ is equal to
If $y=y(x)$ is the solution curve of the differential equation $\frac{dy}{dx}+y\mathrm{tan}x=x\mathrm{sec}x,0\leq x\leq \frac{\pi }{3},y(0)=1$, then $y(\frac{\pi }{6})$ is equal to
If ${a}_{\alpha }$ is the greatest term in the sequence ${a}_{n}=\frac{{n}^{3}}{{n}^{4}+147},n=1,2,3....,$ then $\alpha$ is equal to $______$
If $A$ is the area in the first quadrant enclosed by the curve $C:2{x}^{2}-y+1=0$, the tangent to $C$ at the point $(1,3)$ and the line $x+y=1$, then the value of $60A$ is................
If [ $t$ denotes the greatest integer $\leq 1$, then the value of $\frac{3(e-1)}{e}{\int }_{1}^{2}{x}^{2}{e}^{[x]+[{x}^{3}]}dx$ is :
If $\int \sqrt{\mathrm{sec}2x-1}dx=\alpha {\mathrm{log}}_{e}|\mathrm{cos}2x+\beta +\sqrt{\mathrm{cos}2x(1+\mathrm{cos}\frac{1}{\beta }x)}|$ $+$ constant, then $\beta -\alpha$ is equal to ______.
If $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying ${\int }_{0}^{\frac{\pi }{2}}f(\mathrm{sin}2x)\mathrm{sin}xdx+\alpha {\int }_{0}^{\frac{\pi }{4}}f(\mathrm{cos}2x)\mathrm{cos}xdx=0$, then the value of $\alpha$ is
If $\alpha >\beta >0$ are the roots of the equation $a{x}^{2}+bx+1=0$, and $\underset{x\rightarrow \frac{1}{\alpha }}{\mathrm{lim}}{(\frac{1-\mathrm{cos}({x}^{2}+bx+a)}{2(1-\alpha x{)}^{2}})}^{\frac{1}{2}}=\frac{1}{k}(\frac{1}{\beta }-\frac{1}{\alpha })$, then $k$ is equal to
If $f(x)={x}^{2}+{g}^{'}(1)x+{g}^{"}(2)$ and $g(x)=f(1){x}^{2}+x{f}^{'}(x)+{f}^{"}(x)$, then the value of $f(4)-g(4)$ is equal to _____ .
If $I(x)=\int {e}^{{\mathrm{sin}}^{2}x}(\mathrm{cos}x\mathrm{sin}2x-\mathrm{sin}x)dx$ and $I(0)=1$, then $I(\frac{\pi }{3})$ is equal to
For $m,n>0$, let $\alpha (m,n)={\int }_{0}^{2}{t}^{m}{(1+3t)}^{n}dt$. If ,$11\alpha (10,6)+18\alpha (11,5)=p{(14)}^{6}$, then $p$ is equal to
For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $\int ({(\frac{x}{e})}^{2x}+{(\frac{e}{x})}^{2x}){\mathrm{log}}_{e}xdx=\frac{1}{\alpha }{(\frac{x}{e})}^{\beta x}-\frac{1}{\gamma }{(\frac{e}{x})}^{\delta x}+C$, where $e=\sum _{n=0}^{\infty }\frac{1}{n!}$ and $C$ is constant of integration, then $\alpha +2\beta +3\gamma -4\delta$ is equal to
The derivative of sin(x²) with respect to x is:
∫₀¹ x² dx is equal to:
Area of the region ${(x,y):{x}^{2}+{(y-2)}^{2}\leq 4,{x}^{2}\geq 2y}$ is
A wire of length $20m$ is to be cut into two pieces. A piece of length ${\ell }_{1}$ is bent to make a square of area ${A}_{1}$ and the other piece of length ${\ell }_{2}$ is made into a circle of area ${A}_{2}$. If $2{A}_{1}+3{A}_{2}$ is minimum then $(\pi {\ell }_{1}):{\ell }_{2}$ is equal to:
A square piece of tin of side $30\mathrm{cm}$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in ${\mathrm{cm}}^{2}$) is equal to
${\int }_{0}^{2}(|2{x}^{2}-3x|+[x-\frac{1}{2}])dx$, where $[t]$ is the greatest integer function, is equal to
$\int \frac{({x}^{2}+1){e}^{x}}{{(x+1)}^{2}}dx=f(x){e}^{x}+C$, where $C$ is a constant, then $\frac{{d}^{3}f}{d{x}^{3}}$ at $x=1$ is equal to